"Acts or threats of political violence have no place" in the US, he added, but he refused to call it an act of domestic terrorism.
His omission of the word has enraged many Americans, who have taken to Twitter to ask how sending explosive devices to a news organisation could not be deemed terrorism.
"Call it what it is Donald Trump: 'Radical right wing terrorism,'" wrote Hill Reporter editor Brian Krassenstein on Twitter.
"CNN, the Clintons, the Obamas, George Soros, etc are not our enemies. Homegrown terrorism IS!"
Hollywood star Samuel L. Jackson was more direct.
"TERRORISM Muthaf**kah, Can You Say it???!!!!!!" he wrote in response to the President's comments on Twitter.
CNN president Jeff Zucker has also now lashed out at Trump, criticising the seriousness of his attacks on media.
"There is a complete lack of understanding at the White House about the seriousness of their continued attacks on the media," he said.
"The President, and especially the White House Press Secretary, should understand their words matter. So far, they have shown no comprehension of that."
A US official said the package to addressed to the suburban New York home where Clinton lives with former President Bill Clinton was also a "functional explosive device". It was intercepted at a facility in Chappaqua, Westchester County.
Another bomb was sent to the home Obama shares with wife Michelle in Washington, DC.
A suspicious package addressed to Democratic California congresswoman Maxine Waters was intercepted at a Congressional mail facility.
Former Democratic National Committee chairwoman Deborah Wasserman Schultz also received a suspicious package — addressed to former Attorney-General Eric Holder and sent to her as the return addressee.
Schultz's office in Sunrise, Florida, was evacuated after reports of a possible bomb threat, with Broward Sheriff's Office using a bomb squad robot to investigate the package.
The NYPD's chief of counter-terrorism John Miller said the spate of suspicious packages began with a pipe bomb found on Tuesday at the compound of billionaire philanthropist George Soros, a major donor to Democratic liberal and causes.
Miller said the devices appeared to have been sent by the same person and all appeared to be crude pipe bombs. The explosive found at CNN was black with wires, with the network's reporter Evan Perez describing the device as "rudimentary but functional".
The targets all appear to be left-wing, Democrat-linked figureheads and organisations. The bombs were intercepted just two weeks before the Midterm elections, which could reshape Congress and act as a referendum on the first two years of Trump's presidency.
Trump earlier retweeted a tweet from Vice-President Mike Pence saying they "condemned the attempted attacks" and that those responsible for the "despicable" actions would be brought to justice.
First lady Melania Trump said she condemned "all forms of threats and violence".
Both Republicans and Democrats have already started using the incidents to attack the other side.
Former Clinton aide Philippe Reines placed the blame on Trump. "With every fibre of your rotten being you've incited & condoned hate," he tweeted. "Never in 643 days have you discouraged people from unleashing the violence inside them. Your very existence requires anger & fear. You need it like normal people need oxygen. This is on you."
Peter Daou, who worked on Clinton's 2008 campaign, said Republicans were responsible for "the destruction of our democracy, tweeting: "YOU OWN THIS."
Bill Mitchell, a pro-Trump radio host, tweeted: "These 'explosive packages' being sent to the #Media and high profile Democrats has Soros astro-turfing written all over it so the media can paint the #GOP as 'the dangerous mob.' Pure BS."
Right-wing commentator John Cardillo said in a now-deleted tweet: "Investigators need to take a serious look at far left groups like #Antifa when investigating the bombs sent to Soros, Obama, and the Clintons. These smell like the false flag tactics of unhinged leftists who know they're losing."
New York Governor Andrew Cuomo said a device had been sent to his office in Manhattan, but it was unrelated to the mail bombs.
"We would not be at all surprised if more devices show up," said Cuomo, a Democrat, at a media briefing. "There is a pattern."
A federal official described all the devices as "very similar in style," "legitimate" but "not sophisticated." The devices are still being analysed, and the white powder has been sent for tests, with the NYPD deploying resources around the city in case of more devices.
The street beside the Columbus Circle building, where CNN is based, went into lockdown after the suspicious package was delivered to the mail room.
One local living in the busy area near the Time Warner Centre told news.com.au that police arrived incredibly quickly — but the frequent bomb scares had left her too afraid to live there.
"It's worrying to live in this area," Nicole Doring, originally from New Jersey and now living near the building in Columbus Circle, told news.com.au. "It's because it's the most highly, densely populated area. I've been living here for around three years now and I'm moving in January.
"It's nice to be around everything and everything's so accessible but it's not worth it."
Workers in the Time Warner Centre said they heard fire alarms and were told to leave the building, before hearing there was a suspicious package. Many were calling and texting family to tell them they were safe.
Walter Rivera, who works at a restaurant in the building, said: "I live in New York so whatever happens. I'm a little scared, I've never seen this before, but I'm good."
Construction workers who were on the roof said they had heard there was a bomb as they left the building, and didn't expect to get back in today.
Australian tourists Simon and Narelle Haydock, from Brisbane, were caught up in the drama as they walked past.
"We were walking along and we saw a bit of commotion and police cars everywhere and black cars with all sorts of lights flashing … and then saw the bomb squad here and all the cordons.
"We're just trying to get some more information."
Yellow police tape was put up across normally heaving streets, with massive crowds and a horde of media gathering as several buildings were evacuated. The sound of traffic was replaced by loud sirens from emergency vehicles. Senior counter-terror officials, the NYPD bomb squad, the Bureau of Alcohol, Tobacco, Firearms and Explosives, police dogs and the military attended the scene.
An emergency alert was sent to New Yorkers' phones asking residents on W 58th Street between Columbus Circle and 8th Avenue to "take shelter immediately". It was followed by another saying the package had been removed.
New York City Mayor Bill de Blasio said: "This clearly is an act of terror attempting to undermine our free press and leaders of this free country through acts of violence."
The Secret Service said neither Clinton nor Obama were at risk because of screening procedures.
"We are fine thanks to the men and women of the Secret Service who intercepted the package addressed to us long before it made its way to our home," Clinton said. "Every day, we are grateful for their service and commitment. And obviously, never more than today. But it is a troubling time, isn't it?"
White House spokeswoman Sarah Huckabee Sanders said in a statement: "These terrorising acts are despicable, and anyone responsible will be held accountable to the fullest extent of the law.
"The United States Secret Service and other law enforcement agencies are investigating and will take all appropriate actions to protect anyone threatened by these cowards."
The bomb at CNN was addressed to former CIA Director John Brennan, who appears on air on other broadcast and cable outlets.
The packages had United State Postal Service stamps on them, the official said.
The Joint Terrorism Task Force is investigating and the FBI said it was working with other law enforcement agencies.
"We will not rest until we stop these hazardous devices from being mailed and bring the individual or individuals to justice," said Bryan Paarmann, the FBI's top counter-terrorism official in New York.
Early reports a suspicious device was also found in a parcel addressed to the White House were revealed to be false. The building that houses the San Diego Union-Tribune newspaper and US Senator Kamala Harris was also evacuated after suspicious packages were found, but police said the package did not contain anything dangerous.
